Study of Fluid Dynamics with Emphasis on Couette Flow and Taylor Vortices

Study of Fluid Dynamics with Emphasis on Couette Flow and Taylor Vortices

Project Report submitted to the

FERGUSSON COLLEGE

In Partial Fulfillment for the Degree of

BACHELOR OF SCIENCE

IN

PHYSICS

BY

ABHAY MADHUSUDAN KARNATAKI

(Exam No. 01038)

T. Y. B. Sc.

DEPARTMENT OF PHYSICS

FERGUSSON COLLEGE,

PUNE.

FERGUSSON COLLEGE, PUNE – 411004.

This is to certify that

Mr. Abhay Madhusudan Karnataki

(Exam No. 01038) has satisfactorily completed the project entitled

“Study of Fluid Dynamics with Emphasis on Couette Flow and Taylor Vortices”

under our guidance in partial fulfillment of Degree of Bachelor of Science (B. Sc.) in Physics as per described by University of Pune during the academic year 2000-2001.

A. G. Banpurkar Dr. Mrs. R. Joshi

External Guide Internal Guide

Prof. D. M. KulkarniProf. P. T. Purandare

Project Incharge Head, Department of Physics, Fergusson College, Pune.

To the people who

shaped my life,

my Father, Mother and

Brother and

my beloved teacher

Prof. M. Prakash…

Contents

Preface / Ii
Acknowledgements / iv
1. What is fluid dynamics? / 1
2. Toolkit of fluid dynamics / 3
3. Fluid dynamics in daily life / 13
4. Canned rolls / 21
4.1 Introduction / 22
4.2 While I was canning the rolls / 22
4.3 Observations / 26
4.4 Theoretical explanations / 27
4.4.1 Argument for inviscid fluids / 27
4.4.2 Plane and circular couette flow / 29
4.4.3 Computer generated trajectory / 34
4.4.4 Linear stability theory / 37
4.5 Other interesting things observed during the project work / 38
4.6 Recent work and further prospects / 42
4.7 Applications / 43
5. Playing with soap films / 45
6. My hero - Sir G. I. Taylor / 58
Conclusions / 61
Appendices:
A. ‘C’ program for the particle trajectories / 62
B. Accreting on stars / 64
C. Fascinating BZ reaction / 68
D. More about Navier Stokes equations / 73
E. Some useful properties of common fluids / 75
Suggestions for further reading / 76

Acknowledgements

I express my sincere thanks to my external guide Shri. Arun Banpurkar for taking out time from his busy schedule for Ph.D. thesis and helping me in all possible ways. Many of the ideas in this project report originated and took a better shape in number of exciting discussions with him. Arun sir essentially taught me how to set up an experiment right from the scratch and how to optimize it for the best results with patience and efforts.

I thank my internal guide Dr. Mrs. R. S. Joshi for always showing me the right direction to proceed and for many important suggestions.

It gives me a great pleasure to thank Dr. A. V. Limaye for helpful discussions at several stages, which enabled me to analyze various situations in minute details.

I take this opportunity to express my thanks to Prof. S. B. Ogale for providing me a strong impetus for hard work and future studies. I thank Dr. K. P. Adhi for raising many important questions and for providing many suggestions. I thank all the members of CLAMP; Dr. S. I. Patil, Dr. Bathe, Narhe, Khandkar, Mandar and Sadakale for making the working environment so wonderful and for their help while performing the experiments.

I thank Dr. Tapas K. Das from IUCAA for explaining me in detail the analysis of the process of accretion of matter on stars and related things.

I thank the Heads of Departments of physics in Pune University and Fergusson College for providing me the required infrastructure facilities. I thank the Center for Non-Linear Dynamics, Texas, for sending me the reprints of some research papers on Taylor vortices. I thank our elderly lab assistant Haribhau for his numerous suggestions while making the apparatus.

I thank my friend Sandeep for many exciting discussions on mathematics and especially for his book “What is mathematics?”, which provided me lot of things on soap films. I thank all my friends from chemistry discipline, Ashutosh, Devayani, Sarita and Mrunalini and teachers from department of Chemistry, Pune University, Dr. Avinash Kumbhar and Dr. P. K. Choudhary for their patient help in flow visualization methods and the fascinating BZ reaction. Special thanks to Ashutosh for always sharing the excitement in science.

Because this project work marks the end of my undergraduate studies and as this work was motivated from my previous studies in Physics, I would like to express my thanks to all those who helped me in the last three years. First of all I would like to thank Desai sir from Exploratory for giving me a free access to the lab facilities there, in these years. I thank all the staff members of Physics department in Fergusson College for making the studies of physics so joyful and exciting. Special thanks to Ogale madam, Alawani madam and Dabhade madam for always encouraging me for extra curricular activities. I also thank my Mathematics teachers, especially Acharya sir, Kulkarni sir for always promoting my interest in Mathematics. I also thank my teachers from Electronics department, especially Bhide sir and Khedkar sir for introducing me to the excitement in the world of Electronics. I would like to thank the Professors in Pune University Physics Department Prof.. P. V. Panat, Prof. A.W. Joshi, Dr. C. V. Dharmadhikari, Dr. Mrs. A. Kshirsagar, Dr. R. K. Pathak for their ever-welcoming nature towards my Physics queries.

My special thanks to Dr. A. D. Gangal from Pune University Physics Department for his guidance for reading extra curricular books and for resolving many mysteries of fluid dynamics as well as other topics in “Feynman Lectures on Physics” in a delightful manner. He introduced me the joy of theoretical physics in the simplest ways.

My special thanks also to Dr. S. V. Dhurandhar for teaching me the concepts of special and general theory of relativity in an elegant manner; which has given me a deep satisfaction of studying physics.

I thank all my seniors who have become my close friends, Deepti, Harshad, Prasad, Anuradha, Aditi, Subhangi, Anamica, Aparajita, Deepanjan and Devraj. They have always shown me the opportunities lying ahead and have provided the glimpses of advanced physics. I thank all my friends in colleges for sharing the joy of doing physics. Special thanks to Priya, Vidyut, Sonal, Shashank, Sourabh, Abhijit, Maitreyi, Sheetal, Sulbha and Rahul for many exciting discussions. I thank my chess friends Nivedita, Ashwini and Udayan among others and my chess teacher Mr. Joseph D’ Souza for keeping the chess player in me alive, which was essential for my good studies after leaving chess as a profession. My special thanks to my friend Sneha for her constant support and encouragement, and to Abhijit and Virendra for always being with me in my adventures.

Last but not the least; I thank my friend Vaibhav for helping me in typing this project report and also for being a “test student”! Some names might have got omitted in this acknowledgement by mistake, and the people I have mentioned here have helped me in ways more than I can describe in words. I thank them all with all my sincerity.

Abhay Karnataki.

Chapter 1

What Is Fluid Dynamics?

We think of fluids as liquids or gases only. But the subject of fluid dynamics deals with a large class of systems, than just liquids and gases e.g. Certain types of glasses can be thought of as a fluid. Why? Because they ‘yield’ under the force of gravity. Glasses of windows in old houses are found to be thicker at bottom than at the top. Because glass can flow! So fluids are those substances which cannot stand the shearing forces.

If you push the upper end of a thick book resting on a table, then its pages tilt – its pages slide upon each other. This is the action of shearing stress. Similarly, layers of fluid slide over each other. Now, sliding involves friction, and this sideways force between two layers is called the Viscous Force. The measure of amount of viscous force is the property of fluids called Viscosity. More the viscosity of the fluid, more is the viscous force. And this viscous force stops the fluid layers from moving indefinitely on each other. But, if the external force is applied for sufficiently long time, the fluid layers have to move.

Thus, we may define fluid as, “A Fluid is matter in a readily distortable form, so that the smallest external unbalanced force on it causes an infinite change of shape, if applied for a time long enough” [1].

Fluid dynamics can be used to model a variety of physical situations. E.g. it can be used to study the rate of accretion of stellar dust on a star moving through dust clouds in a galaxy. Now, such dust particles may be a few kilometers apart, but on astronomical scales, we can still treat the dust cloud as a ‘Continuous Fluid’. Fluid dynamics may also be used to model traffic on Mumbai Pune Highway. Here the particles would be of different types, corresponding to various vehicles. But as a whole, they can be treated as a fluid.

But for all purposes, we may study fluid dynamics with the working fluid as water, to start with. Water can be used as a standard example of fluid while studying fluid dynamics.

Having understood what a fluid is, we should try to see what are the various physical parameters associated with the flows of the fluid. It is the wide diversity of these parameters that makes the subject of fluid dynamics so challenging and difficult to understand.

Simplest parameters we can think of are, density of fluid and velocity of fluid at each point inside the fluid. Also, from Bernoulli’s Equation and Hydrostatistics we are familiar with pressure at a point in fluid, which is another important parameter. In addition there can be external forces imposed by the experimenter using pumps, or the forces like gravity, which are omnipresent. On a free surface of a fluid, e.g. on the surface of flow in river, surface tension will be an important factor. To complicate the matter, there may be charges present in the fluid e.g. ions of some chemicals. Moving charges constitute current. So if external electric and magnetic fields are applied, the charges will interact with them. Also, internally the charges will interact with each other. This will modify the flow of the fluid. These branches of study are called ‘Magnetohydrodynamics’, and ‘Electrohydrodynamics’. To further complicate the things, there may be temperature gradient inside the fluid, which will in turn induce density variations in the fluid. There may be density variations produced due to impinging sound waves [2].

Fluid Dynamics is the study of fluid under these various conditions put separately or together. It also deals with interaction of fluid with solid boundaries near it, e.g. body of a fish swimming in a pool of water.

This may give an impression that Fluid Dynamics is a hopelessly complicated subject. But in many real life situations only few parameters are of prime importance. And over the centuries, physicists, with their unique insight into natural phenomena, have discovered many rules that regulate the fluids.

References:

1. A textbook of fluid dynamics- Fransis.

2. The Feyman Lectures on Physics -Vol. II

Chapter 2

Toolkit of Fluid Dynamics

So, how do we formulate a theory of such a nasty fluid, flowing with its own will - so to say, interacting with so many physical parameters?

The central and starting assumption of fluid dynamics is that we can describe the fluid under the consideration as a continuous medium. This is the so-called “Continuum Hypothesis ”.

But we know that there is even in case of water, a lot of void space between two molecules of water. So how can we possibly treat water or any other practical fluid as continuous?

The point is that we will be observing phenomena, which are sufficiently macroscopic in nature. And ‘sufficiently macroscopic’ means that the characteristic length scales over which the flow is examined, are larger than the mean free path of the molecular collisions. E.g. we might be interested in knowing how the presence of an obstacle like an airplane causes change in the flow velocity of the air near it. Now, these changes occur with typical length scales of a few meters, and mean free path of air molecules may be of a few millimeters. Or we may put some obstacle like cylinder in flowing water. Here flow patterns change over a few millimeters, but the mean free path of water molecules is a few micrometers. So, the fluid in consideration can be treated as continuous. Similarly, the dust particles accreting on stars, which are a few kilometers apart, as in case of upper atmosphere of earth, can be treated as continuous medium over the length scales of tens of lakhs of kilometers.

So, when I say that "consider a fluid element of volume dV", I mean a cube of length greater than the mean free path of fluid under consideration, although mathematically we treat dV as an infinitesimal element.

When I say that "a fluid particle" has moved from A to B, I mean the ensemble of particles in a fluid element of volume dV situated at A has moved, on an average, to B. Or the " fluid particle" has a velocity V at (x, y, z), I mean the ensemble of particles in a fluid element of volume dV at (x, y, z) has, on an average, the velocityV.

The particles in that fluid element may have various different velocities and some of the molecules may not go from A to B or some of the molecules may not have velocityV, but the averaged out quantities are important. Same reasoning applies when we say that a layer of fluid moves over another layer.

The continuum hypothesis and the results derived based on it have been tested in the laboratories in various experiments. And they have been proved to be successful since centuries. But we should be aware that it has certain limits imposed by length scales under consideration. In the days of nanotechnology, fluid flowing through nanotubes may not be treated as continuous, and molecular size effects will have to be taken under consideration.

Having made the Continuum Hypothesis, we now describe the equation, which expresses conservation of mass. The mass is conserved means that the change in mass contained in a closed surface per unit time is equal to the total mass that goes in or comes out of the same surface.The component of velocity perpendicular to the surface determines the flux of mass coming in or going out of a surface. The component parallel to the surface just causes the mass to move on the surface. So we find the mass flux through a closed surface ‘S’ as:

Where I have written -the density of the fluid, instead of mass ‘m’, because we will be considering physical quantities per unit volume. This is because the “fluid ‘particle” itself is of a finite volume and not a point particle, as explained previously.

The rate of change of mass enclosed by the same surface is where V is the volume enclosed by the surface. Conservation of mass implies that

….. (1)

where negative sign is introduced to account for the fact that outward flux is considered positive. So, when  decreases with time, i.e. fluid goes out of the surface ‘S’, the flux remains positive.

Left-hand side of (1) can be written as:

,

by Gauss’s divergence theorem. And so, as the volume ‘V’ enclosed by the surface ‘S’ becomes very small, we can write equation 1 as in differential form as:

..… (2)

Equation (2) is called as the ‘Continuity Equation’. This equation is completely analogous to the continuity equation in electrodynamics. There the roll of mass is played by charges and the velocity charges gives rise to currents. But, with essentially the same reasoning, there it expresses the conservation of charges. The continuity equation in this most general form is required for the treatment of the gas coming out of a small nozzle, under high pressure. In that case the gas expands, so density variations become important. But in most of the cases of liquids, we will consider the density of the fluid to be constant. E.g. density of water can be taken to be constant unless, sound waves are passed through it. So, liquids like water, glycerin, paraffin oil etc. will be treated as incompressible. Then the equation of continuity takes a simple form

…..(3)

Equation (3) says that the amount of fluid entering a closed surface is same as that leaving the surface. Or the flux of the fluid i.e. surface integral of is zero. So essentially there are no sinks in the region of concern.

So the continuity equation puts some restrictions on the velocity distribution in the fluid. But, what governs how the fluid moves? The fluid moves under the action of forces like pressure, gravity etc. Thus, 'Newton's second law' decides how the fluid moves.

Let me first show you how the Newton’s second law is written in case of fluids, and then explain the significance of each term.

.….(4)

These vectorial equations are called as “Navier Stokes equations”.

The sum of all forces acting on a fluid particle is written as the right hand side, and the left-hand side of the equation gives acceleration of the particle. Let’s see why. The first term on the right hand side of equation 4 is the force due to pressure changes, (either caused by the motion of the fluid itself or applied externally) in the fluid. Let us consider component in the X-direction of . It is . Consider a layer of fluid perpendicular to x axis as shown below:

Figure 2.1: Pressure forces on a thin fluid layer perpendicular to x-axis. P is the new pressure at x +x.

The force in X-direction on a fluid layer of thickness x is ,where = Area of cross section of the layer perpendicular to X – axis. So, the net force per unit volume acting along the X – axis is: . Hence, the force per unit volume along X –axis is. In vectorial notation, is the force per unit volume in X – direction. Similarly, and are the forces per unit volume in Y & Z – direction.